LEARNING MIXTURES OF RANDOM UTILITY MODELSZhibing Zhao, Tristan Villamil and Lirong Xia

Computer Science Department, Rensselaer Polytechnic Institute

RANK AGGREGATION

• Alternatives: {Donald Trump, Hillary Clinton,Gary Johnson}.

• Mechanism: plurality rule.• Decision: Donald Trump.

RANKING MODELS

• Set of alternatives: A = {a1, a2, . . . , am}.• Parameter space: Θ.• Sample space: i.i.d. rankings over A.• Distributions: {Prθ : θ ∈ Θ}.

RANDOM UTILITY MODELS (RUMS)• Sample a random utility for each alternative independently.• Rank the alternatives w.r.t. these utilities.• The Plackett-Luce model: all utility distributions are Gumbel distributions.

MIXTURES OF RANDOM UTILITY MODELS (k-RUMS)• Parameter: mixing coefficients (α1, α2, . . . , αk); RUM components: RUM1, RUM2, . . ., RUMk.• Sample a component r w.r.t. mixing coefficients.• Sample a ranking from component r.

MOTIVATION: MODEL FITNESS ON PREFLIB DATA

Given the number of parameters d, the number ofrankings n and the likelihood of the estimate L:• AIC = 2d− 2 ln(L).• AICc = AIC + 2d(d+1)

n−d−1 .• BIC = d ln(n)− 2 ln(L).

Observation:

k-RUM � k-PL � RUM � PL,

where A � B means that the number of datasetswhere A beats B is more than that where B beats A.

THEORETICAL RESULTS: IDENTIFIABILITY

Def: For all ~θ1, ~θ2 ∈ Θ, Pr~θ1 = Pr~θ1 ⇒~θ1 = ~θ2.

Theorem 1. Let M be any symmetric RUM from thelocation family. When m ≤ 2k − 1, k-RUMM over malternatives is non-identifiable.

Figure 1: The label switching problem

Theorem 2. For any RUMM where all utility distributions have support (−∞,∞), when m ≥ max{4k − 2, 6},k-RUMM over m alternatives is generically identifiable.

FIRST ALGORITHMS TO LEARN k-RUMGeneralizd Method of Moments (GMM).

1. Choose q events: all pairwise comparisons andselected triple-wise comparisons.

2. Compute the empirical probabilities b1, . . . , bq .3. Find the parameter that minimize

∑qi=1(bi −

pi(~θ))2, where pi(~θ) is the probability computed

from the model.

E-GMM.

1. E-step: compute the probabilities that a rankingbelongs to each component.

2. M-step: compute the parameter of each compo-nent using the GMM algorithm by Azari Soufiani(2014).

The Sandwich Algorithm (GMM-E-GMM).

1. Run GMM for a good starting point.2. Run E-GMM to improve accuracy.

Figure 2: The sandwich algorithm

EXPERIMENTS

Figure 3: 2-RUM over 6 alternatives Figure 4: 4-RUM over 15 alternatives

COMPARISON WITH k-PLS

k-PLs k-RUMsClosed-form likelihood? Yes NoProving identifiability? Hard Harder

REFERENCES

Hossein Azari Soufiani, David C. Parkes, and LirongXia, “Computing Parametric Ranking Models viaRank-Breaking". ICML-14.Zhibing Zhao, Peter Piech, and Lirong Xia, “LearningMixtures of Plackett-Luce Models". ICML-16